Potential at a General Point Due to a Dipole

Q. An electric dipole is situated in an electric field of uniform intensity E whose dipole moment is p and moment of inertia is I. If the dipole is displaced slightly from the equilibrium position, then the angular frequency of its oscillations is

1. ${\left(\frac{pE}{I}\right)}^{1/2}$
2. ${\left(\frac{pE}{I}\right)}^{3/2}$
3. ${\left(\frac{I}{pE}\right)}^{1/2}$
4. ${\left(\frac{p}{IE}\right)}^{1/2}$

Q. When charge of $3$ coulomb is placed in a Uniform electric field, it experiences a force of $3000$ newton, within this field, potential difference between two points separated by a distance of $1\text{cm}$ is (in volt)

Q. Two charges of $-4\mu \text{C}$ and $+4\mu \text{C}$ are placed at points $\text{A}(1,0,4)$ and $\text{B}(2,-1,5)$ located in an electric field $\overrightarrow{E}=0.20\hat{i}{\text{Vcm}}^{-1}$. The torque acting on the dipole is

1. $8\times {10}^{-5}\text{Nm}$
2. $\frac{8}{\sqrt{2}}\times {10}^{-5}\text{Nm}$
3. $8\sqrt{2}\times {10}^{-5}\text{Nm}$
4. $2\sqrt{2}\times {10}^{-5}\text{Nm}$

Q. Find the electric dipole moment due to two charges kept at end points of hypotenuse of a right-angled triangle as shown in figure.

1. $15\text{C-m}$ along $\text{AB}$
2. $20\text{C-m}$ along $\text{AC}$
3. $25\text{C-m}$ along $\text{BC}$
4. $25\text{C-m}$ along $\text{CB}$

Q. A dipole having dipole moment $p$ is in front of a solid uncharged conducting sphere as shown in the figure. The net potential at point A lying on the surface of the sphere is: (where, $K=\frac{1}{4\pi {\epsilon }_0}$)

1. $\frac{Kpco{s}^2\varphi }{r^2}$
2. zero
3. $\frac{2Kpco{s}^2\varphi }{r^2}$
4. $\frac{Kpcos\varphi }{r^2}$

Q. A point dipole $\overrightarrow{p}=-p_0\hat{x}$ kept at the origin. The potential and electric field due to this dipole on the $y-$axis at a distance $d$ are, respectively :
(Take $V=0$ at infinity)

1. $0,\frac{|\overrightarrow{p}|}{4\pi {\epsilon }_0{d}^2}$
2. $0,\frac{-\overrightarrow{p}}{4\pi {\epsilon }_0{d}^3}$
3. $\frac{|\overrightarrow{p}|}{4\pi {\epsilon }_0{d}^2},\frac{\overrightarrow{p}}{4\pi {\epsilon }_0{d}^3}$
4. $0,\frac{\overrightarrow{p}}{4\pi {\epsilon }_0{d}^3}$

Q. An electric field of $1000\text{V/m}$, is applied to an electric dipole moment of ${10}^{-29}C\cdot m.$ What is the potential energy of the electric dipole?

1. $-20\times {10}^{-18}J$
2. $-7\times {10}^{-27}J$
3. $-10\times {10}^{-29}J$
4. $-9\times {10}^{-20}J$

Q. As shown in the figure, on bringing a charge $Q$ from point $\text{A}$ to $\text{B}$ and from $\text{B}$ to $\text{C}$, the work done are $2\text{Joules}$ and $-3\text{Joules}$ respectively. The work done in bringing the charge from $\text{C}$ to $\text{A}$ will be

1. $-1\text{Joule}$
2. $1\text{Joule}$
3. $2\text{Joules}$
4. $5\text{Joules}$

Q.

A 4 $\mu$F capacitor is given 20 $\mu$C charge and is connected with an uncharged capacitor of capacitance 2 $\mu$F as shown in figure. When switch S is closed –

1. Charged flown through the battery is $\frac{40}{3}\mu C$

2. Charge flown through the battery is $\frac{20}{3}\mu C$

3. Work done by the battery is $\frac{200}{3}\mu J$

4. Work done by the battery is $\frac{100}{3}\mu J$

Q. Two small electric dipoles each of dipole moment $p$ are situated at $(0,0,0)$ and $(r,0,0)$. The electric potential at a point $(\frac{r}{2},\frac{\sqrt{3}r}{2},0)$ is

1. $\frac{p}{4\pi {ϵ}_0{r}^2}$
2. $0$
3. $\frac{p}{2\pi {ϵ}_0{r}^2}$
4. $\frac{p}{8\pi {ϵ}_0{r}^2}$

Q. Two charges $-4\mu \text{C}$ and $+4\mu \text{C}$ are placed at the points $\text{A}(1,0,4)$ and $\text{B}(2,-1,5)$ located in an uniform electric field $\overrightarrow{E}=20\hat{i}\text{N/C}$. The torque acting on the dipole has a magnitude of

1. $0.1\times {10}^{-4}\text{Nm}$
2. $1.13\times {10}^{-4}\text{Nm}$
3. $2.5\times {10}^{-4}\text{Nm}$
4. $3.7\times {10}^{-4}\text{Nm}$

Q. A wire carrying a current of $3\text{A}$ is bent in the form of a parabola $y^2=4-x$ as shown in figure, where $x$ and $y$ are in metre. The wire is placed in a uniform magnetic field $\overrightarrow{B}=5\hat{k}\text{T}$. The magnetic force acting on the wire is:

1. $\left(60\hat{i}\right)\text{N}$
2. $(-60\hat{i})\text{N}$
3. $\left(30\hat{i}\right)\text{N}$
4. $(-30\hat{i})\text{N}$

Q. The figure given below shows four orientations of an electric dipole in an external electric field.

1. Potential energy of dipole is greater for orientations $\left(1\right)$ and $\left(3\right)$ than $\left(2\right)$ and $\left(4\right)$.
2. If the dipole rotates from orientation $\left(1\right)$ to $\left(2\right)$, then the workdone on the dipole by field is positive.
3. If the dipole rotates from orientation $\left(1\right)$ to $\left(2\right)$, then the workdone on the dipole by field is negative.
4. Workdone to rotate dipole from $\left(1\right)$ to $\left(2\right)$ is same as workdone to rotate dipole from $\left(1\right)$ and $\left(4\right)$.

Q. There is a fixed positive charge $Q$ at $O$. $A$ and $B$ are points equidistant from $O$. A positive charge $+q$ is taken slowly by an external agent from $A$ to $B$ along $ACB$. Then,

1. The total work done on the charge is zero.
   
2. The work done by the electrostatic force from $A$ to $C$ is negative.
   
3. The work done by the electrostatic force from $C$ to $B$ is positive.
   
4. The work done by electrostatic force in taking the charge from $A$ to $B$ is dependent on the actual path.
   

Q. What should be the orientation of an electric dipole in a uniform electric field, that corresponds to stable equilibrium ?

1. $\theta ={90}^{\circ }$
2. $\theta ={180}^{\circ }$
3. $\theta ={30}^{\circ }$
4. $\theta =0^{\circ }$

Q. Find the electric dipole moment due to two charges kept at end points of hypotenuse of a right-angled triangle as shown in figure.

1. $15\text{C-m}$ along $\text{AB}$
2. $20\text{C-m}$ along $\text{AC}$
3. $25\text{C-m}$ along $\text{BC}$
4. $25\text{C-m}$ along $\text{CB}$

Q. Two equal charges q of opposite sign separated by a distance 2a constitute an electric dipole of dipole moment p . If P is a point at a distance rfrom the centre of the dipole and the line joining the centre of the dipole to this point makes an angle $\theta$ with the axis of the dipole, then the potential at Pis given by (r >> 2a) (Where p = 2qa)

1. $V=\frac{pcos\theta }{4\pi {ϵ}_0{r}^2}$
2. $V=\frac{pcos\theta }{4\pi {ϵ}_{0}r}$
3. $V=\frac{psin\theta }{4\pi {ϵ}_{0}r}$
4. $V=\frac{pcos\theta }{2\pi {ϵ}_0{r}^2}$

Q. A thick conductor having inner radius $a$ and outer radius $b$ has a small passage as shown in the figure.


A charge $+Q$ is placed at the centre. The work done by external agent in taking the charge from centre to infinity is

1. $\frac{Q^2}{8\pi {\epsilon }_{0}a}-\frac{Q^2}{8\pi {\epsilon }_{0}b}$
2. $-\frac{Q^2}{4\pi {\epsilon }_{0}a}-\frac{Q^2}{4\pi {\epsilon }_{0}b}+\frac{Q^2}{4\pi {\epsilon }_0(a-b)}$
3. $-\frac{Q^2}{8\pi {\epsilon }_{0}b}$
4. $\frac{Q^2}{8\pi {\epsilon }_{0}a}$

Q. The electric dipole potential at a large distance $r$ from the dipole falls off. The falling off the magnitude of the electric potential depends on,

1. $\frac{1}{r}$
2. $\frac{1}{r^4}$
3. $\frac{1}{r^3}$
4. $\frac{1}{r^2}$

Q. Two point charges $+q$ and $-q$ are held fixed at $(-a,0)$ and $(a,0)$ respectively of a $x-y$ coordinate system, then

1. the electric field $\overrightarrow{E}$ at all points on the $x-$ axis has same direction
2. $\overrightarrow{E}$ at all points on the $y-$ axis is along $\hat{i}$
3. $\overrightarrow{E}$ at all points on the $y-$ axis is along $-\hat{i}$
4. $\overrightarrow{E}$ at all points on the $y-$ axis is along $\hat{j}$

Q.
Current I is flowing through two materials having electrical conductivities ${\sigma }_1$ and ${\sigma }_2$ where ${\sigma }_1>{\sigma }_2$, then total amount of charge at the junction of materials is

1. zero
2. positive
3. negative
4. may be positive or negative

Q. An electric dipole of moment $\overrightarrow{p}=(\hat{i}-3\hat{j}+2\hat{k})\times {10}^{-29}\text{Cm}$, is at the origin. The electric field due to this dipole at $\overrightarrow{r}=+\hat{i}+3\hat{j}+5\hat{k}$ (note that $\overrightarrow{r}\cdot \overrightarrow{p}=0)$ is parallel to:

1. $(+\hat{i}-3\hat{j}-2\hat{k})$
2. $(-\hat{i}+3\hat{j}-2\hat{k})$
3. $(+\hat{i}+3\hat{j}-2\hat{k})$
4. $(-\hat{i}-3\hat{j}+2\hat{k})$

Q. Two charges of equal magnitude 'q' but of opposite sign separated by a distance 2a constitute an electric dipole of dipole moment p. If P is a point at a distance r from the centre of the dipole and the line joining the centre of the dipole to this point makes an angle $\theta$ with the axis of the dipole, then the potential at the point P is given by $(r\gg 2a)$ (Where p=2qa )

1. $V=\frac{pcos\theta }{4\pi {ϵ}_0{r}^2}$
   
2. $V=\frac{pcos\theta }{4\pi {ϵ}_{0}r}$
   
3. $V=\frac{psin\theta }{4\pi {ϵ}_{0}r}$
   
4. $V=\frac{pcos\theta }{2\pi {ϵ}_0{r}^2}$

Q. Derive an expression for electric potential at a point due to an electric dipole. Explain the special cases.

Q. An electric dipole of dipole moment $p$ is placed in a uniform electric field $E$ in stable equilibrium position. Its moment of inertia about the centroidal axis is $I$. If it is displaced slightly from its mean position, find the period of small oscillations.

1. $2\pi \sqrt{\frac{I}{pE}}$
2. $\frac{1}{2\pi }\sqrt{\frac{pE}{I}}$
3. $\pi \sqrt{\frac{I}{pE}}$
4. $2\pi \sqrt{\frac{IE}{p}}$

Q. Which of the following is a property of equipotential surfaces?

$[$0.77 Mark$]$

1. The direction of electrostatic field will be perpendicular to the equipotential surface
2. The potential difference between any two points on an equipotential surface is zero
3. For a point charge, the equipotential surfaces will be concentric spheres
4. All the statements above are correct.

Q. A spherical metal shell A of radius $R_A$ and a solid metal sphere $B$ of radius $R_B(<R_A)$ are kept far apart and each is given '$+Q$'. Now they are connected by thin metal wire. Then:

1. $E_A^{inside}=0$
2. $Q_A>Q_B$
3. $\frac{{\sigma }_A}{{\sigma }_B}=\frac{R_B}{R_A}$
4. $E_A^{onsurface}<E_B^{onsurface}$

Q. An electric dipole is situated in an electric field of uniform intensity E whose dipole moment is p and moment of inertia is I. If the dipole is displaced then the angular frequency of its oscillation is :

1. $(\frac{pE}{I}{)}^{\frac{1}{2}}$
2. $(\frac{pE}{I}{)}^{\frac{3}{2}}$
3. $(\frac{I}{pE}{)}^{\frac{1}{2}}$
4. $(\frac{p}{IE}{)}^{\frac{1}{2}}$

Q. Two short electric dipoles are placed as shown ($r$ is distance between their centres). The energy of electric interaction between these dipoles will be:
($C$ is centre of dipole of moment $P_2$)

1. $\frac{2k{P}_1{P}_2\cos \theta }{r^3}$
2. $-\frac{2k{P}_1{P}_2\cos \theta }{r^3}$
3. $\frac{-2k{P}_1{P}_2\sin \theta }{r^3}$
4. $\frac{-4k{P}_1{P}_2\cos \theta }{r^3}$

Q.

This section contains 1 Matrix Match type question, which has 2 Columns (Column I and Column II). Column I has four entries (A), (B), (C) and (D), Column II has four entries (P), (Q), (R) and (S). Match the entries in Column I with the entries in Column II. Each entry in Column I may match with one or more entries in Column II.

इस खण्ड में 1 मैट्रिक्स मिलान प्रकार का प्रश्न है, जिसमें 2 कॉलम (कॉलम I तथा कॉलम II) हैं। कॉलम I में चार प्रविष्टियाँ (A), (B), (C) तथा (D) हैं, कॉलम II में चार प्रविष्टियाँ (P), (Q), (R) तथा (S) हैं। कॉलम I में दी गयी प्रविष्टियों का मिलान कॉलम II में दी गयी प्रविष्टियों के साथ कीजिए। कॉलम I में दी गयी प्रत्येक प्रविष्टि का मिलान कॉलम II में दी गयी एक या अधिक प्रविष्टियों के साथ हो सकता है।

Match the Columns and choose the appropriate answer

कॉलमों का मिलान करें और सही उत्तर चुनें

Consider the following electrostatic field lines due to a positive point charge. Four points A, B, C & D are also shown in the figure.

एक धनात्मक बिन्दु आवेश के कारण नीचे दी गई स्थिरवैद्युत क्षेत्र रेखाओं पर विचार कीजिए। चित्र में चार बिन्दु A, B, C व D भी दर्शाए गए हैं।




Match the entries given in Column-I with information given in Column-II.

कॉलम-I में दी गई प्रविष्टियों का मिलान कॉलम-II में दी गई सूचना के साथ कीजिए।

	Column I कॉलम I		Column II कॉलम II
(A)	At A, as compared to B B की तुलना में A पर	(P)	Potential is high विभव उच्च है
(B)	At C, as compared to A A की तुलना में C पर	(Q)	Potential is low विभव निम्न है
(C)	At B, as compared to C C की तुलना में B पर	(R)	Magnitude of electric field is low विद्युत क्षेत्र का परिमाण निम्न है
(D)	At D, as compared to B B की तुलना में D पर	(S)	Magnitude of electric field is high विद्युत क्षेत्र का परिमाण उच्च है

1. A-PS; B-QR; C-PS; D-QR
2. A-P; B-Q; C-PS; D-QR
3. A-S; B-Q; C-PS; D-R
4. A-PQS; B-PQR; C-PS; D-QRS